Compact Conical Line Power Combiner Design
Using Circuit Models
Ryno D. Beyers, Student Member, IEEE, and Dirk I. L. de Villiers, Member, IEEE
Abstract—A simple equivalent circuit model with empirical equations describing the peripheral feeding ports of conical line power combiners is presented. The model allows the entire structure to be designed using transverse electromagnetic circuit theory without the need for any full-wave simulations. A sum-mary of the model extraction process is given and the accuracy of the proposed model is confirmed by favorable comparisons with full-wave simulations. The circuit based design method is used to design a compact conical line combiner showing measured performance similar to the current state of the art combiners in this technology, while being significantly smaller.
Index Terms—Circuit models, combiners, conical combiners, conical transmission lines, N -way splitters, optimization, passive components, radial combiners.
I. INTRODUCTION
A
XIALLY symmetric N -way power combiners offer a number of advantages over conventional corporate and chain combiners when N is large. These advantages include higher combining efficiencies due to reduced insertion loss and improved amplitude and phase balance, as well as re-duced physical size and weight [1]–[3]. With few exceptions, these types of combiners have traditionally been difficult to design: Electromagnetic field analysis is used in [4], empirical techniques based on measurements is used in [5], and for some others no detailed design information is provided [6]– [8]. With the advent and continued improvement of 3D elec-tromagnetic modeling software, the analyses and simplified design approaches of many variations based on radial, coaxial and conical transmission lines have been presented [9]–[15].The conical transmission line implementation of N -way power combiners [10] is a relatively new technology offering some advantages over the more conventional coaxial line [12], [16] and radial line [9] structures. The conical combiners in [10] and [11] are designed using a hybrid technique where a minimal number of full-wave simulations are required: Even though conical transmission lines support a fundamental trans-verse electromagnetic (TEM) mode and may thus be designed by circuit theory, the peripheral ports of conical combiners that transition into the conical lines contain discontinuities where higher order modes are excited, and cannot be modeled by simple TEM transmission lines [17]. Equivalent circuit models have previously been used to model different parts – including in some cases the peripheral port transitions – of various types of combiners. However, most of these circuit models do not offer a means to relate the circuit element values The authors are with the Department of Electrical and Electronic Engi-neering, University of Stellenbosch, Stellenbosch 7600, South Africa (email: 15407756@sun.ac.za; ddv@sun.ac.za).
to physical dimensions [12], [18], or they are not accurate enough to be used exclusively and full-wave optimization or parameter sweeps are still needed afterwards to obtain the final dimensions of the structures [9]–[11], [13], [14].
This paper presents a set of general empirical equations based on full-wave simulations that describe the circuit model, as presented in [19] and shown in Fig. 1, for shorted coaxial peripheral feeding ports in conical combiners. The empirical equations allow the designer to determine the equivalent circuit element values accurately and directly from the physical dimensions of conical combiners and vice versa. In many cases the empirical equations are accurate enough to allow the circuit model to be used exclusively during the design process, eliminating the need for full-wave analyses. This allows for rapid optimization of various dimensions of the combiner at a significantly reduced computational cost compared to full-wave optimization together with matching networks that may be required for wide band operation. This method also enables the designer to minimize the total transmission length and thus the physical size of the combiner.
The parameter extraction method and the subsequent model derivation are discussed, followed by some example designs using the circuit model. Full-wave simulations of the final structures are performed to confirm the accuracy of the pro-posed method, where close correlations between the circuit models and the full-wave results are obtained. The proposed circuit based design method is validated by comparing the simulation results of an example design with its measurements.
II. PHYSICALDESCRIPTIONANDEMPIRICALEQUATION EXTRACTION
The basic layout of a conical combiner is shown in Fig. 1, where the different regions that will be used in the circuit model description are indicated by dotted boxes. Note that the figure is rotationally symmetric around the vertical axis on the left. The equivalent circuit model for the combiner and the location of external matching networks that may be added is shown in Fig. 2, where the equivalent circuit for each region of the combiner is contained within its corresponding box. The peripheral feeding port transition used here is the uncompensated version without tuning posts, as defined in [19], and is different from what is used in [10] and [11]. Note that the circuit model is only valid for the symmetrically driven case where the fields at the peripheral input ports have the same amplitude and phase. In Fig. 1, regions A and F are coaxial lines, regions C and E are conical lines, and region B is a constant impedance conical to coaxial line
Fig. 1. Cut plane view of the basic layout of a conical line combiner showing the different regions used in the circuit model.
Fig. 2. Circuit model of the full combiner showing the regions corresponding to the physical model in Fig. 1, including external matching networks that may be needed.
Fig. 3. A smooth conical to coaxial transmission line transition, as presented in [20].
transition, as shown in Fig. 3. These regions are all simple TEM transmission lines and can thus be modeled by ideal transmission lines with lengths and impedances derived from the physical geometry of the structure, whereas region D includes some reactive elements to compensate for the stored evanescent mode energy around the peripheral coaxial feeding port to conical line transition. Note that the transmission line in region F and the inductor in region D represent a parallel combination ofN of those components for an N -way combiner.
A complete and accurate physical description of the com-bining structure is needed so that the transmission line lengths and impedances needed for the circuit model extraction and the
circuit model based design procedures can be calculated. The equations needed for regions A, B, C, E and F will be given, followed by the extraction process for the empirical equations needed for region D. Note that no external matching networks are used during the extraction process.
A. Central Output Coaxial Line (Region A)
Region A contains a constant impedance coaxial line with inner and outer conductor radii of R1 and R2, respectively.
The coaxial line in region A will have the same length lA as
used in the ideal transmission line model. The impedanceZA
can be calculated using
ZA= 60ln
R2
R1
. (1)
B. Central Transition (Region B)
The central transition from conical to coaxial line is de-signed using the smooth transition presented in [20] and shown in Fig. 3. The radii of the two arcs (r1andr2) used to construct
the transition are obtained using
r1= 3.5× (R2− R1) , (2)
r2=
(R1+ r1) cosθ1B
1− cosθ1B
, (3)
whereθ1B is the conical line angle as defined in Fig. 3. The
mean transmission length (the dashed line in region B, Fig. 1) can be calculated using
lB=
r1+ r2
2 ×
θ2B+ θ1B
2 , (4)
whereθ2B is the conical line angle as defined in Fig. 3, and
usuallyθ2B = 90◦. The impedance of the transition is shown
to be constant in [20] and can be determined by calculating either the coaxial or conical transmission line impedances:
ZB= ZA= 60ln R2 R1 , (5) or ZB= 60ln cot(θ1B/2) cot(θ2B/2) (6) for air-filled coaxial and conical lines, respectively.
C. Conical Transmission Line (Region C)
The mean transmission length for this region is calculated from the edge separating regions D and C to the edge separating regions C and B. The length (the dashed line in region C, Fig. 1) can be calculated using
lC= rp cos(π/4− θ1D/2)− ln− ds /2, (7) where ln= 1 2[R1+ R2+ r1+ r2(1− cosθ1B)] (8) is the average length of conical transmission line removed from the central part of the conical line where the transition to the central coaxial line is inserted. In (7)rp is the peripheral
Fig. 4. A section of the combining structure showing the parts of the top conductor being removed where the coaxial lines are placed, as well as the port numbering used throughout this work.
Fig. 5. The constant impedance 10-way conical combiner used to extract the model. The shaded part of the model is vacuum and the background material is perfect electrical conductor (PEC).
input port placement radius as defined in Fig. 4, ds is the
arithmetic mean ofdsin Fig. 4 as the lineOP is moved from
the positionOP1toOP2and can be approximated using (13),
and the same definition as in region B is used forθ1Dexcept
that it is the angle of the conical line in region D. The length lCis exact when region C is a constant impedance conical line,
whereθ1B= θ1D, and is a reasonable approximation if region
C is an impedance tapered conical line, where θ1B6= θ1D. If
region C is a constant impedance conical line, thenZC= ZB,
since θ1B = θ1D and θ2B = θ2D = 90◦. If region C is an
impedance tapered conical line then the conical line angleθ1C
required to realize the desired impedance function Zf versus
distance can be approximated as a function of radial distance from the axis of symmetry of the conical lineρ, using
θ1C(ρ) = 2arctan
tan(θ2B/2)
eZf(ρ−ln)/60
. (9)
The impedance functionZf can be an impedance taper of the
designer’s choice, such as an exponential or a Hecken [21] taper.
D. Conical Transmission Line (Region E)
The length (the dashed line in region E, Fig. 1) can be calculated using
lE=
rb
cos(π/4− θ1D/2)− ds/2,
(10)
where rb is the back-short length as defined in Fig. 4. The
impedanceZE can be calculated using
ZE = 60ln cot(θ1D/2) cot(θ2D/2) , (11) sinceθ1D= θ1E andθ2D= θ2E.
E. Coaxial Transmission Line (region F)
Region F contains a constant impedance coaxial line with an inner conductor radiusrinner and an outer conductor diameter
ofdc. The coaxial line in region F will have the same lengthlF
as used in the ideal transmission line model. The impedance ZF can be calculated using
ZF = 60ln d c 2rinner . (12)
F. Empirical Equation Extraction (region D)
The length lD in the circuit model is calculated directly
from the combiner dimensions, whereas empirical equations will be used to calculate the impedance ZD and inductance
LD. The empirical equations are extracted so that they are
applicable to a wide range of conical line combiners. A sector of anN -way combiner is shown in Fig. 4 where the dimensions used to derive the expression for the length lD are defined. The outer conductors of the peripheral ports
are formed by drilling holes through one of the conical transmission line conductors and are represented by the circles between the dashed arcs T and V.
The drilling of the holes results in the removal of some of the conical line conductor, causing a change in impedance be-tween arcs T and V. The change in impedance is approximated using two short transmission lines. It is assumed that rp is
large relative to the outer conductor diameter of the peripheral coaxial lines dc, so that the arc U will be approximately
straight inside the removed circle area. As a result, the average of ds and thus the lengthlD, can be approximated simply by
using
lD= ds=
dcπ
4 . (13)
The value ofZDis calculated by scaling the impedance of
the conical transmission line to model the effect of removing some of the conductor from the conical line. This modification is expected to increase the impedance of the conical line between arcs T and V, resulting in a scaling factor for ZD
that is larger than1. Furthermore, the value of ZDis expected
to be dependent on the ratio of the amount of conductor along the circumference of the conical line at radius rp before the
holes are made to the amount remaining after the holes are made. The empirical equation forZD can thus be defined as
ZD= g1(x1)Zsys, (14) where x1= 2πrp 2πrp− Nds = rp rp−N d8c , (15)
Zsys is the impedance of the unperturbed conical line
TABLE I
EXTRACTEDPOLYNOMIALSFORTHEEMPIRICALEQUATIONS
Function Units
g1(x1, ∆r) = −0.054x1∆r + 0.48x1+ 0.072∆r + 0.38 Ω/Ω
g2(x2, ∆r) = 62x2∆r + 320x2− 230∆r − 5.7 pH
(Zsys = ZE), and N is the number of peripheral feeding
ports of the combiner.
The inductor in region D models the extended center conductor pin of the peripheral coaxial lines and its value is expected to be dependent on the pin length. Thus,
LD= g2(x2), (16)
where
x2= rpcotθ1D (17)
is the length of the coaxial pin extending into the conical transmission line.
A simple constant impedance conical combiner as shown in Fig. 5 is used to extract the empirical equations that describe ZD and LD. This is done by fitting the scattering
parameters of the circuit model onto the corresponding ones produced by full-wave simulations. All full-wave simulations are performed using the time domain solver in Computer Simulation Technology (CST) Microwave Studio (MWS) [22]. A mean square error function is defined in order to measure how well the model matches the full-wave simulations:
ε = K X k=1 1 K|S f 11(fk)− S11c (fk)|2 . (18) Sf11andSc
11are the S-Parameters of the full-wave and circuit
simulations, respectively, and fk is thekth frequency sample
of a total of K samples. Port 1 is the central port and ports 2 to N + 1 are the peripheral ports, as defined in Fig. 4. A Nelder-Meade based Simplex search [23] is used to minimize ε over a wide bandwidth (> 100%) around a chosen center frequency by adjusting the values ofZDandLDin the circuit
model. The center frequency is determined by the length of the back-short in the conical line rb (also defined in Fig. 4),
which is equal to a quarter wavelength at that frequency. A center frequency of 10 GHz is used to extract the empirical equations. Straight lines fit the resulting values ofZDandLD
well, however, the coefficients of the best fitting lines change for different peripheral port dimensions. A straight line is thus fitted to each set of data points sharing the same peripheral port dimensions, resulting in a number of different equations. These equations are combined by fitting polynomials to the obtained coefficients for different peripheral port dimensions versus ∆r, with
∆r = dc/2− rinner, (19)
whererinner is the inner conductor radius, anddc is the outer
conductor diameter of the peripheral coaxial feeding ports. The resulting empirical equations are the simple bivariate polynomials given in Table I. Note that all dimensions should be specified in millimeters. 0.0025 0.005 0.01 0.01 0 Zs y s N /Z F (rpπ)/(rbN ) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
(a) 10 GHz (X-Band) combiners with 3.5 mm peripheral ports.
0.0025 0.005 0.01 0.02 0.01 Zs y s N /Z F (rpπ)/(rbN ) 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1
(b) 10 GHz (X-Band) combiners with 85.6 Ω peripheral ports with SubMinia-ture version A (SMA) connector inner conductor dimensions.
0.0025 0.005 0.01 0.02 0.02 Zs y s N /Z F (rpπ)/(rbN ) 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.6 0.7 0.8 0.9 1 1.1 1.2
(c) 10 GHz (X-Band) combiners with N-type peripheral ports.
0.0025 0.005 0.01 0.02 Zs y s N /Z F (rpπ)/(rbN ) 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.6 0.7 0.8 0.9 1 1.1 1.2
(d) 6 GHz (C-Band) combiners with N-type peripheral ports. Fig. 6. Contour plots of the mean square error, as defined in (18), between full-wave simulations and their equivalent circuit models. The accuracy of the circuit models used for the designs in section V are indicated by ×-markers on the contour plots: The 30 port X-Band combiner is shown in (a), the 10 port X-Band combiner in (b) and the 15 port C-Band combiner in (d).
III. PARAMETERSTUDY
A parameter study is performed to test the accuracy of the model. Equivalent circuit models for various combiners are built using the information presented in Section II (a detailed design procedure is given in Section IV). Combiners with center frequencies of6 GHz (C-Band) and 10 GHz (X-Band) and peripheral feeding ports with the same inner and outer conductor radii as the standard 50 Ω 3.5 mm and N-type connectors are used to generate the data shown in Fig. 6. A combiner with 85.6 Ω peripheral feeding ports with the same inner conductor radius as the standard SubMiniature version A (SMA) connector is also used, similar to what is used in [11], with a different outer conductor radius from the standard 50 Ω SMA connector. The quantities Zsys,ZF,N , rp, andrb
in Fig. 6 are defined in Section II.
The error function defined in (18) is used to show how well the scattering parameters of the circuits match their corresponding full-wave simulations for different combiner dimensions over a larger than100% bandwidth. These contour plots may be of interest to the designer when using the model. The need for full-wave simulations can be eliminated by limiting the combiner dimensions to regions whereε is small, however this model could still serve well as a coarse model for space mapping techniques in situations where the combiner dimensions cannot be limited to the higher accuracy regions, or when the results are not satisfactory.
Fig. 6 has been generated using data for10-way combiners. However, similar data for15-way and 20-way combiners with 85.6 Ω SMA feeding ports has been generated and compared to the data forN = 10. A statistical analysis of the data reveals that the model accuracy is relatively independent of N : The mean error function variance for the three values of N is
Var(ε)mean= 3.19× 10−7, (20)
and the maximum error function variance is
Var(ε)max= 1.42× 10−5. (21)
A comparison of the results for combiners with N-type ports operating at X-Band [Fig. 6(c)], wheredc/rb≈ 0.9, and at
C-Band [Fig. 6(d)], wheredc/rb≈ 0.5, shows that the electrical
size of the peripheral ports influences the accuracy of the cir-cuit model: The region with higher accuracy (ε < 2.5× 10−3)
is much larger at C-Band than at X-Band. The same effect can be seen by comparing Figs. 6(a) and 6(c), which are for combiners with the same center frequency, but with different peripheral port sizes. As a general rule, the outer conductor diameter of the peripheral ports should be less than a quarter of a wavelength at the center frequency, thus dc/rb < 1, and
increased accuracy is expected for smaller diameters. IV. DESIGNPROCEDURE
The physical description, equivalent circuit model, and empirical equations presented in Section II are used to compile a step-by-step design procedure so that the designer may use the presented information in a systematic way. The number of input ports N , the wavelength at the center frequency of the operating band λc, and the type of connectors to be used
for the peripheral input ports and central output port should be selected before starting the design procedure. The first part of the design consists of setting up initial values and/or constraints for the parameters that can be optimized. The optimizable parameters are all either physical dimensions or can be directly related to physical dimensions of the combiner. The parameter study performed in Section III is used to create a set of recommendations that will assist the designer in obtaining more accurate results. The second part of the design consists of analyzing and optimizing the equivalent circuit model for one or more design goals. When satisfactory results are obtained, a 3D model of the combiner may be constructed and a single analysis performed using full-wave simulation software, such as CST MWS, in order to verify the design. A. Recommended initial values or constraints for optimizable parameters
1) For the back-short lengthrb, it is recommended that
rb≈
λc
4 . (22)
2) For the outer conductor diameter dc, of the peripheral
input coaxial lines it is recommended that
dc< rb. (23)
Consider that the coaxial lines modeled by region F need to interface with matching networks or – if the matching networks are omitted – directly with the input connectors. Calculate the resulting values for ZF using
dc and the radius of the inner conductorrinnerthat will
be used for the peripheral input ports using (12). 3) It is recommended, in general, that for the impedance of
the unperturbed conical line in region D, as described in Section II,
Zsys≈
ZF
N . (24)
4) For the peripheral input port placement radius rp,
im-proved accuracy of the circuit model can be obtained, in general, when
rp< N rb
π , (25)
while also keepingrp large enough to accommodate all
of the input connectors for ports2 to N + 1.
5) For the outer conductor radius R2 of the coaxial line
in region A, the fact that this coaxial line will need to interface either directly with the chosen output connector or with an output matching network should be taken into consideration.
6) The inner conductor radius R1 of the coaxial line
in region A also affects the conical to coaxial line transition in region B, since ZA = ZB. Furthermore,
if region C is a constant impedance conical line, then ZA = ZB = ZC = Zsys = ZE. However, using
a constant impedance conical line in region C may lead to manufacturing difficulties and inaccuracies due to the small spacing (R2 − R1) that is required in
order to realize a low impedance coaxial transmission line, as pointed out in [11]. It is thus recommended
to use a tapered conical line in region C that tapers the impedance fromZsys up to a higher impedance in
regions A and B, resulting in a larger spacing and thus improved manufacturability. In this caseR1is chosen or
optimized and constrained to provide adequate spacing, andrp will affect the taper length.
B. Calculation of equivalent circuit model element values The equivalent circuit model of the entire combining struc-ture, as shown in Fig. 2, can now be constructed in a circuit simulator. Fig. 2 shows the location of external matching networks, for example stepped impedance coaxial lines, that may be added by the designer. The designer will need to find or derive the equivalent circuit models for any added external matching networks needed for a specific combiner design. The circuit element values of the equivalent model can now be calculated using the parameters described in Section IV-A as variables, by using the following procedure:
1) CalculateZB from (5).
2) Calculateθ1B using (6) where typically θ2B= 90◦.
3) lB can be calculated using (4).
4) lA can be optimized together with an external output
matching network, or if region A is already matched to the desired output port impedance and dimensions, lA
can be zero. 5) ZA= ZB.
6) lC can be calculated using (7).
7) If region C is a constant impedance conical line then ZC = ZB, otherwise the profile of the
de-sired impedance taper, such as an exponential or a Hecken [21] taper, with a length of lC should be
calculated using (9). 8) Calculate∆r using (19). 9) lD can be calculated using (13).
10) ZD can be calculated by combining (14), (15), and the
functiong1(x1, ∆r) listed in Table I.
11) Calculateθ1D using (11) where typicallyθ2D= 90◦.
12) LD can be calculated by combining (16), (17), and the
functiong2(x1, ∆r) listed in Table I.
13) lE can be calculated using (10).
14) ZE= Zsys, and thus withθ2E = θ2D,θ1E= θ1D.
15) lF can be optimized together with an external input
matching network, or if region F is already matched to the desired input port impedance and dimensions,lF
can be zero.
16) ZF can be calculated using (12).
The entire circuit model including external matching networks and the impedance tapered conical line in region C can now be optimized for one or more chosen design goals.
V. DESIGN EXAMPLES
The circuit model is further validated by completing some example designs with external input and output matching net-works and comparing the S-parameters of the model with the full-wave simulations. Three different combiners are designed with stepped impedance central coaxial ports to match them to 50 Ω, similar to the combiner in [10], using [24] to calculate
(a) Frequency (GHz) |S1 1 | (d B ) −30 −20 −10 0 4 6 8 10 12 14 16 18 Circuit Model Full Wave (b)
Fig. 7. (a) The full-wave simulation model of the 30-way X-Band combiner with standard 3.5 mm connector dimension peripheral ports, and (b) the comparison between the full-wave and equivalent circuit model output port reflection coefficients (S11).
the step capacitances. In each of the combiners, the impedance of the conical lines are tapered up to higher values near the central port, as is done in [11], except that a smooth Hecken taper [21] is used instead of a Klopfenstein taper [25]. These examples also serve as an indication of how well the circuit model S-Parameters match the full-wave simulations for combiners that fall into different accuracy regions as shown in Fig. 6 and explained in section III.
The first design is for an X-Band 30-way combiner, with a center frequency of 10 GHz, that has 50 Ω peripheral ports with the same inner conductor radius as the standard3.5 mm connector. The 3D model used for the full-wave simulation is shown in Fig. 7(a), and excellent agreement between the circuit model and full-wave simulation is shown in Fig. 7(b). This level of accuracy is achieved by limiting the combiner dimensions to the higher accuracy and thus lower error regions as indicated by the×-marker in Fig. 6(a).
The second design is for a C-Band 15-way combiner, shown in Fig. 8(a), with a center frequency of 6 GHz and 50 Ω peripheral ports with inner conductor radii corresponding to the standard N-type connector dimensions. A comparison between the circuit model and full-wave simulation results is shown in Fig. 8(b), with slightly deteriorated but still good agreement considering that this combiner falls into a much lower accuracy region [see Fig. 6(d)] compared to the previous design. This example demonstrates that the model is valid for a different frequency range.
The third design is for an X-Band 10-way combiner, with a center frequency of 10 GHz, that has stepped impedance peripheral ports with a constant inner conductor radius equal to that of the standard SMA connector. The peripheral ports are stepped into a65.4 Ω partially filled coaxial transmission line followed by a 85.6 Ω section that transitions into the conical transmission line, as shown in Fig. 9(a). This is similar
(a) Frequency (GHz) |S1 1 | (d B ) −30 −20 −10 0 0 2 4 6 8 10 12 Circuit Model Full Wave (b)
Fig. 8. (a) The full-wave simulation model of the 15-way X-Band combiner with standard N-type connector dimension peripheral ports, and (b) the comparison between the full-wave and equivalent circuit model output port reflection coefficients (S11).
to the peripheral ports used in [11]. The stepped impedance feeding lines add degrees of freedom, namely the lengths of the 65.4 Ω (lpar) and85.6 Ω (lF) lines, that can be optimized. The
impedance step introduces a small shunt capacitance that can be omitted due to its small effect. The central port reflection coefficient (S11) is shown in Fig. 9(b) and the circuit model is
in excellent agreement with the full-wave simulation. For this designrp + rb = 25.9 mm compared to rp + rb = 40 mm
in [11], while exhibiting similar performance. The reduction in size is mainly due the fact that the impedance taper in the conical line no longer needs to be designed as in [11], where the taper length is maximized in order to achieve the best possible reflection coefficient in the passband. This was required since the combining structure, and thus the taper, was not included in the optimization parameter space since full-wave analysis was used to find the response. The circuit model approach used here allows the taper to be optimized to-gether with the impedance levels and transmission line lengths throughout the entire combiner, and it can consequently have a shorter length. The final parameters of the optimized combiner are: R2 = 3.5 mm, ZA = ZB = 20.18 Ω, lA = 0,
ZE = Zsys = 9 Ω, dc = 5.164 mm, rinner = 0.62 mm,
rp = 17 mm, rb = 7.9 mm, ZF = 85.6 Ω, lF = 9.5 mm,
lpar = 4 mm. The stepped coaxial output matching network
has impedance levels of32.89 Ω and 38.62 Ω, and lengths of 4.4 mm and 4.2 mm, in that order, followed by a 50 Ω coaxial line. The Hecken taper in region C has B = 0 + j2.47, withB defined in [21]. This design is chosen for construction and measurement.
VI. CONSTRUCTION ANDMEASUREMENTS
A computer numerically controled (CNC) lathe is usually able to machine conical structures, such as conical
transmis-(a) Frequency (GHz) |S1 1 | (d B ) −50 −40 −30 −20 −10 0 4 6 8 10 12 14 16 18 Circuit Model Full Wave Full Wave Mod Measured Full Wave Meas
(b)
Fig. 9. (a) The full-wave simulation model of the 10-way X-Band combiner with standard SMA connector dimension peripheral ports, and (b) the compar-ison between the circuit model, full-wave, modified full-wave and measured output port reflection coefficients (S11). The full-wave simulated S11 using
the measured profile of the manufactured combiner (shown in Fig. 10) is also shown in (b).
sion lines, with ease. There are, however, a few limitations that need to be considered. The finite radius of the cutting tool tip limits the size of the smallest concave feature of the structure. The tool tip radius is taken into account by blending all concave corners with a radius equal to or larger than the tip radius. For this design, this modification has very little effect on the combiner performance, since the tip radius (0.4 mm in this case) is much smaller than the guided wavelength at X-Band (λg ≈ 30 mm). Additionally, all the areas in
the combiner requiring this modification have relatively low local field intensities, further reducing its effect. For full-wave simulation purposes, the impedance taper in the conical line is defined by a series of coordinates connected by short straight lines. For construction, the impedance taper is much more conveniently defined by a series of tangential circle sections passing through or near the series of coordinates. The full-wave simulation results of the modified combiner in Fig. 9(b) show that while these modifications significantly reduce the manufacturing effort, the combiner performance is barely affected at all.
The size, shape, and angle of the cutting tool holder and/or toolpost that is used imposes limitations on the realizable shape of the structure. The goal is to use the least number of different cutting tools, since each interchanging of tools increases the cost and introduces a degree of uncertainty, as well as visible and often palpable step discontinuities. If necessary, it is desirable to change the cutting tool at a large radius in this type of structure, since any discontinuities or uncertainties will have less of an effect where the energy is spatially more dispersed. The shape and angle of the cutting tool also influences the amount of effort needed during
ρ (mm) He igh t (m m ) −8 −6 −4 −2 0 2 4 2 4 6 8 10 12 14 Measured Designed (a) ρ (mm) He igh t (m m ) −2 0 2 4 14 16 18 20 22 24 26 28 Measured Designed (b) (c)
Fig. 10. The measured profile of the manufactured device is compared to the CAD model dimensions in (a) and (b), and a photo of the manufactured top and bottom halves of the combiner is shown in (c).
fabrication and whether a certain shape is realisable at all. For example, if the structure has a profile that does not increase or decrease monotonically in height versus radius, as is the case with the chosen design example, the cutting tool needs to be sufficiently narrow and its holder appropriately shaped so that it has enough clearance of the rest of the structure at all times.
The profile of the machined part is measured and compared to the 3D CAD model dimensions in Fig. 10 showing excellent agreement with the design. The largest errors can be seen in the coaxial to conical transition and the impedance taper in the conical line. A photo of the manufactured top and bottom halves of the combiner is shown in Fig. 10(c).
The measured central port reflection coefficient and periph-eral port isolation are shown in Figs. 9(b) and 11 and are in good agreement with their simulated values. A full-wave
Frequency (GHz) Is ol at ion (d B ) 0 5 10 15 20 7 8 9 10 11 12 13 S32 S42 S52 S62 S72 (a)
Fig. 11. The full-wave simulated isolation (dashed lines) compared to the measured isolation (solid lines) of the combiner in its operating band.
Frequency (GHz) 6S n 1 (d eg ) −180 −90 0 90 180 7 8 9 10 11 12 13 |Sn 1 |( d B ) −11 −10 −9 (a) Frequency (GHz) Los se s (d B ) 0 0.2 0.4 0.6 7 8 9 10 11 12 13 (b)
Fig. 12. The measured phase and amplitude balance is shown in (a), where n is the peripheral port number with n = 2, ..., N + 1. The total insertion loss of the combiner is shown in (b).
TABLE II
COMPARISONWITHOTHERRECENTWORK
Ref. Type N ReturnLoss (dB) Bandwidth Frequency Band
This Work Conical 10 18 46% X-Band
[11] Conical 10 18.5 47% X-Band [10] Conical 10 14.7 74% X-Band [9] Radial 30 14 15% Ku-Band [15] Radial 10 15 35% Ku-Band [12] Coaxial 8 12 112% L-Band [13] Coaxial 10 15 30% Ku-Band
simulation is performed using the measured dimensions of the manufactured combiner and the resulting central port reflection coefficient is shown in Fig. 9(b). The remaining difference between the measured and simulated S11 could be due to a
number of factors, such as the SMA to N-type adapter or the non-ideal SMA terminations used during the measurements.
The central port return loss and fractional bandwidth is shown in Table II for comparison with other work. The mea-sured peripheral port isolation is better than6 dB compared to roughly6 dB in [11] and 5 dB in [10]. The measured amplitude
and phase balance is shown in Fig. 12(a). The maximum measured amplitude and phase imbalance is ± 0.6 dB and ± 3◦, respectively, versus ± 0.7 dB and ± 5◦ in [11], and
± 1.5 dB and ± 10◦ in [10]. The insertion loss shown in
Fig. 12(b) is calculated by substituting the measured values for Sj1,j = 2, 3, ..., N + 1 into Losses =−10log10 N +1 X j=2 |Sj1|2 . (26)
The maximum insertion loss in the operating band is0.28 dB, which is the same as in [11], and an improvement compared to [10], where a stepped impedance matching network is used.
VII. CONCLUSION
A simple equivalent circuit model has been presented to-gether with empirical equations that allow for rapid circuit based design and optimization of conical power combiners with shorted coaxial feed ports. The results of a parametric study on the accuracy of the circuit model are presented in a format that may be helpful to the designer. The effectiveness of the circuit model has been demonstrated by using it to design a significantly smaller combiner with performance comparable to previously published designs. The manufactured design exhibits excellent agreement with the circuit model and full-wave simulations.
ACKNOWLEDGMENT
The authors would like to thank Comar International and Reutech Radar Systems (Pty) Ltd. in Stellenbosch, South Africa for assistance in manufacturing and financial support of the project.
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Ryno D. Beyers (S’13) was born in Bellville, South Africa, on April 22, 1989. He received the B.Eng degree in electrical and electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa, in 2011, and is currently working towards the Ph.D. degree in electrical and electronic engineering at the University of Stellenbosch.
His current research interests include passive de-vices and network synthesis.
Dirk I. L. de Villiers (S’05-M’08) was born in Langebaan, South Africa, on October 13, 1982. He received the B.Eng and Ph.D. degrees in electrical and electronic engineering from the University of Stellenbosch, Stellenbosch, South Africa in 2004 and 2007 respectively. During 2005 to 2007 he spent several months as visiting researcher with the Computational Modeling and Programming group at the University of Antwerp in Antwerp, Belgium.
From 2008 to 2009 he was a post-doctoral fellow at the University of Stellenbosch working on antenna feeds for the South African SKA program. During this time he was also a part time lecturer at the Cape Peninsula University of Technology. He is currently a senior lecturer at the University of Stellenbosch, and his main research interests include reflector antennas as well as the design of wide band microwave components such as combiners, filters, and antennas.
